Sound deadening is good, but overkill for some applications, IMHO. Certainly we could tear out the entire interior to apply sound deadening material on every interior surface, but that's a bit much for most.
I really don't want my LBC to sound like a new-er car and enjoy the "noise". Most Triumphs are <span style="font-style: italic">supposed</span> to rattle a bit, since the body is bolted to the frame! However, some LBCs suffer from excessive cockpit heat (Healeys; GT6s; Spits, etc.). In these cars, DynaMat or a non-flammable sound deadening material is useful.
Use a modern material that's specific for the job: read the label, do NOT introduce anything flammable in your cockpit - this is why it pays to purchase a material that's specifically designed for automotive sound deadening, such as DynaMat, etc.
On my GT6s, I apply DynaMat to the entire inside surface of the gearbox tunnel cover with really good results. Lining the tunnel cover renders a dramatic reduction in cockpit heat. Also, applying it to the firewall and floor pans helps reduce heat and noise, as does sealing the gearshft lever hole off. Sealing the gearshift lever hole also keeps fumes from entering the cockpit, a <span style="text-decoration: underline">big</span> improvement in safety and comfort, I've found.
Just be sure to use a fire-resistant, approved material.